The Stacks Project


Tag 0CDC

Chapter 51: Semistable Reduction > Section 51.14: Semistable reduction

Example 51.14.1. Let $R$ be a discrete valuation ring with uniformizer $\pi$. Given $n \geq 0$, consider the ring map $$ R \longrightarrow A = R[x, y]/(xy - \pi^n) $$ Set $X = \mathop{\rm Spec}(A)$ and $S = \mathop{\rm Spec}(R)$. If $n = 0$, then $X \to S$ is smooth. For all $n$ the morphism $X \to S$ is at-worst-nodal of relative dimension $1$ as defined in Algebraic Curves, Section 49.20. If $n = 1$, then $X$ is regular, but if $n > 1$, then $X$ is not regular as $(x, y)$ no longer generate the maximal ideal $\mathfrak m = (\pi, x, y)$. To ameliorate the situation in case $n > 1$ we consider the blowup $b : X' \to X$ of $X$ in $\mathfrak m$. See Divisors, Section 30.29. By construction $X'$ is covered by three affine pieces corresponding to the blowup algebras $A[\frac{\mathfrak m}{\pi}]$, $A[\frac{\mathfrak m}{x}]$, and $A[\frac{\mathfrak m}{y}]$.

The algebra $A[\frac{\mathfrak m}{\pi}]$ has generators $x' = x/\pi$ and $y' = y/\pi$ and $x'y' = \pi^{n - 2}$. Thus this part of $X'$ is the spectrum of $R[x', y'](x'y' - \pi^{n - 2})$.

The algebra $A[\frac{\mathfrak m}{x}]$ has generators $x$, $u = \pi/x$ subject to the relation $xu - \pi$. Note that this ring contains $y/x = \pi^n/x^2 = u^2\pi^{n - 2}$. Thus this part of $X'$ is regular.

By symmetry the case of the algebra $A[\frac{\mathfrak m}{y}]$ is the same as the case of $A[\frac{\mathfrak m}{y}]$.

Thus we see that $X' \to S$ is at-worst-nodal of relative dimension $1$ and that $X'$ is regular, except for one point which has an affine open neighbourhood exactly as above but with $n$ replaced by $n - 2$. Using induction on $n$ we conclude that there is a sequence of blowing ups in closed points $$ X_{\lfloor n/2 \rfloor} \to \ldots \to X_1 \to X_0 = X $$ such that $X_{\lfloor n/2 \rfloor} \to S$ is at-worst-nodal of relative dimension $1$ and $X_{\lfloor n/2 \rfloor}$ is regular.

    The code snippet corresponding to this tag is a part of the file models.tex and is located in lines 5838–5887 (see updates for more information).

    \begin{example}
    \label{example-blowup}
    Let $R$ be a discrete valuation ring with uniformizer $\pi$.
    Given $n \geq 0$, consider the ring map
    $$
    R \longrightarrow A = R[x, y]/(xy - \pi^n)
    $$
    Set $X = \Spec(A)$ and $S = \Spec(R)$.
    If $n = 0$, then $X \to S$ is smooth.
    For all $n$ the morphism $X \to S$ is at-worst-nodal
    of relative dimension $1$ as defined in
    Algebraic Curves, Section \ref{curves-section-families-nodal}.
    If $n = 1$, then $X$ is regular, but if $n > 1$, then $X$ is not
    regular as $(x, y)$ no longer generate the maximal ideal
    $\mathfrak m = (\pi, x, y)$. To ameliorate the situation
    in case $n > 1$ we
    consider the blowup $b : X' \to X$ of $X$ in $\mathfrak m$.
    See Divisors, Section \ref{divisors-section-blowing-up}.
    By construction $X'$ is covered by three affine pieces
    corresponding to the blowup algebras $A[\frac{\mathfrak m}{\pi}]$,
    $A[\frac{\mathfrak m}{x}]$, and $A[\frac{\mathfrak m}{y}]$.
    
    \medskip\noindent
    The algebra $A[\frac{\mathfrak m}{\pi}]$ has generators
    $x' = x/\pi$ and $y' = y/\pi$ and $x'y' = \pi^{n - 2}$.
    Thus this part of $X'$ is the spectrum of $R[x', y'](x'y' - \pi^{n - 2})$.
    
    \medskip\noindent
    The algebra $A[\frac{\mathfrak m}{x}]$ has generators $x$,
    $u = \pi/x$ subject to the relation $xu - \pi$. Note that this ring
    contains $y/x = \pi^n/x^2 = u^2\pi^{n - 2}$. Thus this part of
    $X'$ is regular.
    
    \medskip\noindent
    By symmetry the case of the algebra $A[\frac{\mathfrak m}{y}]$ is
    the same as the case of $A[\frac{\mathfrak m}{y}]$.
    
    \medskip\noindent
    Thus we see that $X' \to S$ is at-worst-nodal of relative dimension $1$
    and that $X'$ is regular, except for one point which has an
    affine open neighbourhood exactly as above but with $n$ replaced by $n - 2$.
    Using induction on $n$ we conclude that there is a sequence of
    blowing ups in closed points
    $$
    X_{\lfloor n/2 \rfloor} \to \ldots \to X_1 \to X_0 = X
    $$
    such that $X_{\lfloor n/2 \rfloor} \to S$ is
    at-worst-nodal of relative dimension $1$ and
    $X_{\lfloor n/2 \rfloor}$ is regular.
    \end{example}

    Comments (2)

    Comment #2444 by Bronson Lim on March 5, 2017 a 5:00 am UTC

    Should say S = \mathop{\rm Spec}(R).

    Comment #2487 by Johan (site) on April 13, 2017 a 10:46 pm UTC

    Thanks, fixed here.

    Add a comment on tag 0CDC

    Your email address will not be published. Required fields are marked.

    In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

    All contributions are licensed under the GNU Free Documentation License.




    In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?